We focus on pair production in time dependent inhomogeneous electric fields given in Eq. (REF ) such that electric fields are inhomogeneous on \(x\) - axis, and the complete DHW equations of motion are reduced to the following 4 differential equations [1]}
\(&D_t \mathbb {s} - 2 p_x \mathbb {p} = 0 , \\&D_t \mathbb {v}_{0} + \partial _{x} \mathbb {v}_{1} = 0 , \\&D_t \mathbb {v}_{1} + \partial _{x} \mathbb {v}_{0} = -2 m \mathbb {p} , \\&D_t \mathbb {p} + 2 p_x \mathbb {s} = 2 m \mathbb {v}_{1} , \) 
